I would add that извинить is often translatable as "to excuse" and простить as "to forgive" -- in prayers, for example, Russian speakers will often say Прости меня, Господи ("Forgive me, O Lord"). So, as people have said, простить can suggest a deeper sense of regret for the wrong act, even though the two words express very similar ideas.

Also, by the way, you can also say (я) прошу прощения -- "I beg forgiveness" -- to express a strong apology. Don't forget that the 1st-person singular present of the verb просить ("to ask; to request") has a ш in the present first-person singular (я прошу, "I am asking"), while the noun прощения (from простить, "to forgive") has a щ. (The 1st-sing future of простить also has a щ: я прощу, "I shall forgive") Also, прощения ends in я because it's in the genitive, not the accusative.